An in-depth numerical study of the two-dimensional Kuramoto–Sivashinsky equation

Abstract

The Kuramoto-Sivashinsky equation in one spatial dimension (1D KSE) is one of the most well-known and well-studied partial differential equations. It exhibits spatio-temporal chaos that emerges through various bifurcations as the domain length increases. There have been several notable analytical studies aimed at understanding how this property extends to the case of two spatial dimensions. In this study, we perform an extensive numerical study of the Kuramoto-Sivashinsky equation (2D KSE) to complement this analytical work. We explore in detail the statistics of chaotic solutions and classify the solutions that arise for domain sizes where the trivial solution is unstable and the long-time dynamics are completely two-dimensional. While we find that many of the features of the 1D KSE, including how the energy scales with system size, carry over to the 2D case, we also note several differences including the various paths to chaos that are not through period doubling.

Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia

“…In what follows we present and analyse the schemes for the one-dimensional system (5.7)-(5.8), noting that the two-dimensional system (5.5)-(5.6) is treated analogously (see also Kalogirou (2014); Kalogirou et al (2015)). where the constant c > 0 is determined later.…”

Nonlinear interfacial instability in two-fluid viscoelastic Couette flow

“…In the present study the elasticity of the lower layer gives ζ > 0 (see (16) that W B e = 0 for Newtonian thick layers), and so there is a competition between elastic instabilities and inertial stabilization when C > 0. In the computations presented in figure 8 we investigate the nonlinear development of the dynamics in such cases.…”

“…Nonlinear simulations of the evolution equation (16) are used to investigate the dynamical states that arise when the flow is linearly unstable. We focus on the influence of the interfacial elastic stresses (and the parameter ζ) and the non-local term which is non-zero when the viscosities of the two fluids are distinct.…”

“…The evolution equation in this case simplifies to the Kuramoto-Sivashinsky (KS) equation which has been studied extensively (e.g. [14,16,30] and references therein). Again, a variety of unsteady nonlinear states are found as ζ is varied as shown in figure 5b.…”

One-dimensional Kardar-Parisi-Zhang and Kuramoto-Sivashinsky universality class: Limit distributions

“…It has been suggested, since the 1980s, that the Kuramoto-Sivashinsky (KS) PDE, a deterministic interface-growth model for a height field h(x, t), which is used in studies of chemical waves, flame fronts, and the surfaces of thin films flowing under gravity [19][20][21][22][23][24][25], is a simplified model for turbulence [23]. It has been conjectured [26], and subsequently shown by compelling numerical studies [27][28][29][30][31][32], in both one dimension (1D) and two dimensions (2D), that the long-distance and long-time behaviors of correlation functions, in the spatiotemporally chaotic NESS of the KS PDE, exhibit the same power-law scaling as their couterparts in the the Kardar-Parisi-Zhang (KPZ) equation [33][34][35][36], a stochastic PDE (SPDE), in which the height field h(x, t) is kinetically roughened. The elucidation of the statistics of h(x, t) in the KPZ SPDE has played a central role in nonequilibrium statis-tical mechanics, in general, and interface-growth phenomena, in particular.…”